Spaces with σ-point finite bases

Abstract

Theorem. Let X be a T₁ space. The following are equivalent:

• (1)

  X has a σ-disjoint base.

• (FX2)

  X is quasi-developable and has a base that is the union of a sequence of rank 1 collections.

• (3)

  X has a quasi-development ($\text{G}$_n) with the property that for each x, $\text{{st}{}^2\text{(x,}\text{G}{}_{\mathrm{n}}\text{): x ∈st}{}^2\text{(x,}\text{G}{}_{\mathrm{n}}\text{)}$, n a positive integer} is a base for $\text{N}$ (x).

Theorem. Let X be a T₁ space. The following are equivalent:

• (1)

  X has a σ-point finite base.

• (2)

  X has a quasi-development ($\text{G}$_n) with each $\text{G}$_n well ranked.

• (3)

  X has a quasi-development ($\text{G}$_n) with each $\text{G}$_n Noetherian of sub-infinite rank.

• (4)

  X has a quasi-development ($\text{G}$_n) with each $\text{G}$_n Noetherian of point finite rank.